Rating-downgrade risk is where the business of the issuer undergoes changing business conditions (or a “shock’) which leads to a downgrading by the rating agency/agencies. This of course leads to an increase in the risk premium on the particular bond.
Risk-premium risk refers to the risk of the overall risk-premium on corporate bonds increasing as a result of changing business conditions generally. For example, in economic recession periods, corporate bond investors may feel that the risk of companies defaulting on principal and/or interest increases. This leads to an increase in the risk premium demanded, which means that the prices of corporate bonds fall.